---
tag: 0D5KR33H7X
kind: definition
name: Baire space
msc: ["54E52"]
defines: 0ZH5AJ2ED7
uses: [0G5AV9Q4RR, 0C38S0KMJK, 0ZG61TFBES]
sources:
  - url: https://en.wikipedia.org/wiki/Baire_space
    locator: Definition
---

## Definition

A [[topological space|0CNQXVBHCX]] $X$ is a Baire space when it has the property that follows.
Let $(F_n)_{n \in \mathbb{N}}$ be a [[countable|0ZG61TFBES]] family of [[closed subsets|0G5AV9Q4RR]] of $X$ with $\operatorname{int} F_n = \emptyset$ for each $n$.
Then also $\operatorname{int} \bigcup_n F_n = \emptyset$.

## Remarks

Here $\operatorname{int}$ is the [[interior|0C38S0KMJK]].
Equivalently, in a Baire space each [[countable intersection of dense open sets is dense|0X2T63GFQQ]].
In a Baire space, each [[meagre set|0FQPYAY9PJ]] has empty interior. Thus a nonempty Baire space is not meagre in itself.
Each complete metric space is a Baire space by the [[Baire category theorem|0A7KGAYHGV]]. The space $\mathbb{Q}$ is not a Baire space.

## Explanations

### L1 human

In a Baire space, countably many thin closed sets can never fill up an open set.
This says that the space is large. It lets you prove that some objects exist: you show that the bad cases form a small set.